What other names does Superman have

Yes, this has happened quite a few times, including adoptions. As far as I know, the only one that has occurred since the New 52 reboot is Jon Lane Kent.

Superboy

The latest incarnation of Superboy in the New 52 is Jon Lane Kent, who is Superman's son from an alternate future. In this series, he's actually more powerful than his father, because he was born under a yellow sun but has his father's Krpytonian genetics. The majority of the story takes place in the future, and Superman features only in flashbacks of an occasional crossover.

I believe he dies in the latest issue of Superboy.

Also, the previous Superboy in the New 52 was a clone of Jon Kent named Kon-El, who was genetically engineered from Kal-El and Lois Lane, arguably making him also their son.

Otherwise

There are literally dozens of other cases where Superman is depicted as having children. Typically, these are single-panel images, showing "Superman's happy ending", and often the children are adopted. In many of these stories, Superman has a child with Wonder Woman, not Lois Lane, presumably on the theory that another superhero would be more likely to.. survive? ... the process.

One significant departure from this is the Elseworlds story Son of Superman:

This story also feature Jon Kent, son of Kal-El and Lois Lane. However, the Elseworlds stories are well outside the mainstream continuity, taking place entirely in their own separate universes. (In this story, for example, Pete Ross and Lana Lang are terrorists, funded by Wonder Woman, and Lex Luthor runs the Justice League.)